A simple method of calculating the age of the earth is based on the natural abundances of ${ } ^ { 235 } \mathrm { U }$ and ${ } ^ { 238 } \mathrm { U }$. It is assumed that the quantities of these two isotopes were equal at the time that the earth formed, and measurements show that the natural abundances at present are $0.72 \% { } ^ { 235 } \mathrm { U }$ and $99.28 \% { } ^ { 238 } \mathrm { U }$. Using the assumptions above and the half-lives of these two isotopes, $7.1 \times 10 ^ { 8 }$ for ${ } ^ { 235 } \mathrm { U }$ and $4.51 \times 10 ^ { 9 }$ years for ${ } ^ { 238 } \mathrm { U }$, calculate the age of the earth.

Question

Quizplus · Accepted Answer

The Answer is At the time the earth was created the ratio of uranium isotopes was
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N _ { 235 } ( 0 ) / N _ { 238 } ( 0 ) = 1.0

and at present it is

N _ { 235 } ( t ) / N _ { 238 } ( t ) = 0.72 / 99.28 = 7.25 imes 10 ^ { - 3 }

The time dependence of the number of nuclei for each isotope can be written as

N _ { 235 } ( t ) = N _ { 235 } ( 0 ) \cdot \exp \left( - \lambda _ { 235 } t ight)

and

N _ { 238 } ( t ) = N _ { 238 } ( 0 ) \cdot \exp \left( - \lambda _ { 238 } t ight)

Taking the ratio of these two expressions (along with the current isotope ratio) gives

\frac { N _ { 235 } ( t ) } { N _ { 238 } ( t ) } = \frac { N _ { 235 } ( 0 ) \cdot \exp \left( - \lambda _ { 235 } t ight) } { N _ { 238 } ( 0 ) \cdot \exp \left( - \lambda _ { 238 } t ight) } = \exp \left[ \left( \lambda _ { 238 } - \lambda _ { 235 } ight) t ight] = 7.25 imes 10 ^ { - 3 }

Solving for

t

(the age of the earth) gives

t = \left[ \ln \left( 7.25 imes 10 ^ { - 3 } ight) ight] / \left( \lambda _ { 238 } - \lambda _ { 235 } ight)

The decay constants of the two isotopes are found from their respective half-lives as

\lambda _ { 235 } = ( \ln 2 ) / au _ { 1 / 2 } = 0.693 / \left( 7.1 imes 10 ^ { 8 } \mathrm { y } ight) = 9.76 imes 10 ^ { - 10 } \mathrm { y } ^ { - 1 }

and

\lambda _ { 238 } = ( \ln 2 ) / au _ { 1 / 2 } = 0.693 / \left( 4.51 imes 10 ^ { 9 } \mathrm { y } ight) = 1.537 imes 10 ^ { - 10 } \mathrm { y } ^ { - 1 }

Substituting in the expression for the age of the earth gives

t = ( - 4.927 ) / \left( - 8.223 imes 10 ^ { - 10 } \mathrm { y } ^ { - 1 } ight) = 5.99 imes 10 ^ { 9 } \mathrm { y }

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A Simple Method of Calculating the Age of the Earth 235U{ } ^ { 235 } \mathrm { U }235U

A Simple Method of Calculating the Age of the Earth ${ } ^ { 235 } \mathrm { U }$